Let's start with the Constitution Article 1, Section 4, Clause 1
The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by Law make or alter such Regulations, except as to the Places of chusing Senators.
Congress has a pretty good page on this
To accomplish the ends under this clause, Congress may adopt the statutes of the states and enforce them by its own sanctions. It may punish a state election officer for violating his duty under a state law governing congressional elections. It may, in short, use its power under this clause, combined with the Necessary and Proper Clause, to regulate the times, places, and manner of electing Members of Congress so as to fully safeguard the integrity of the process; it may not, however, under this clause, provide different qualifications for electors than those provided by the states.
Under US v Classic, Congress can regulate primaries as well.
It's unclear where that line stops, though. Courts haven't restricted the Federal government from doing too many things. Generally states can still
- Draw their own district lines
- Set times for primary elections
- Establish the rules for voting (subject to the Voting Rights Act). This includes rules like registration, days to vote, places to vote, etc.